Cubic identity for three variables — lesson. Mathematics State Board, Class 9.

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Identity:

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

Example:

Expand \(27x^3 + 8y^3 + z^3 - 18xyz\).

Solution:

Let us write the expression of \(27x^3 + 8y^3 + z^3 - 18xyz\) using the identity

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

{27 x}^{3} + {8 y}^{3} + z^{3} - 12 xyz = {(3 x)}^{3} + {(2 y)}^{3} + {(z)}^{3} - 3 (3 x) (2 y) (z)

\(=\)

(3 x + 2 y + z) ({9x}^{2} + {4y}^{2} + z^{2} - (3x) (2y) - (2y) (z) - (3 x) (z))

\(=\)

(3 x + 2 y + z) ({3x}^{2} + {2y}^{2} + z^{2} - 6 xy - 2 yz - 3 xz)

Important!

a^{3} + b^{3} + c^{3} = 0

then the identity

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

is rewritten as follows:

a^{3} + b^{3} + c^{3} - 3 abc = (0) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

a^{3} + b^{3} + c^{3} - 3 abc = (0)

a^{3} + b^{3} + c^{3} = 3 abc

Example:

Evaluate \(8^3 - 5^3 - 3^3\).

Solution:

By the identity if

a^{3} + b^{3} + c^{3} = 0

, then

a^{3} + b^{3} + c^{3} = 3 abc

On comparing \(8^3 - 5^3 - 3^3\) with

a^{3} + b^{3} + c^{3} = 0

we have \(a\) \(=\) \(8\), \(b\) \(=\) \(5\) and \(c\) \(=\) \(3\).

Here \(a+b+c\) \(=\) \(8-5-3\) \(=\) \(0\)

Therefore, \(8^3 - 5^3 - 3^3\) \(=\) \(3 \times 8 \times 5 \times 3\) \(=\) \(360\)

Did you find an error?

4. Cubic identity for three variables